Elements of Physical Biology

able number of drafts, say 50 double drafts, in which a record is kept of all the numbers drawn, the urns are opened, and the tickets in the second urn are now blackened. The drawing is then continued, for, say, another 50 drafts, recording each time the numbers drawn. ‘The numbers on the tickets enable us to trace the previous history of the 50 black tickets, before they were blackened. In an experiment actually carried out it was found that these 50 black tickets were originally distributed essentially evenly in the two urns. The curve representing the first 50 drafts is an ascending curve, the system passed, during this stage of the process, from more probable to less probable states, as shown in the first, ascending portion of the more heavily drawn curve in figure 1. In the second series of fifty drafts the curve descends in normal fashion, with increasing probability of the successive states of the system. It may seem like a contradiction of terms that what amounts practically to an infinitely improbable series of drafts should be capable of actual realization at will. But if the series of drafts described were extended to great length in both directions, say one million drafts before blackening the tickets, and one million after, it would be seen that the peak on the curve is indeed a very exceptional feature. It is a perfectly safe bet that in two million drafts not more than one such peak, going up to 50 black balls in one urn, would be encountered.

The model described exemplifies among others the fact that in an exceedingly long lapse of time it may some time occur that the system will return to its original state. This is quite in accord with the laws of mechanics; in fact, as already noted, these laws actually demand that every mechanical system of finite dimensions must ultimately return to its initial state, and must do this not once only, but in everlasting reiteration at regular intervals: the motion is periodic. This property also is capable of illustration by a simple model, such as the following: Twenty-six pendulums of periods T’ = 0.5, 0.6, . . . 2.9, 3.0 seconds are started simultaneously to the left from their equilibrium position, and are then allowed to oscillate undisturbed. Count is then made, at the end of every tenth of a second, of the number of pendulums on the left of the median. In this way the staircase curve figure 2 was obtained (computation here taking the place of actual observation). It will be seen that in the brief fragment of a period covered by the

record, this exhibits all the characteristics of a “passage from a less probable to a more probable distribution,” though, in point of fact, we know that the system has a perfectly definite period of 7385 years. The appearance of chance in this wholly determinate mechanical process is brought into still greater prominence if we plot the deviations, from the mean, of the number of pendulums on the left of the median position, at successive counts. We thus obtain the points indicated by small circles in figure 3. These group themselves very obviously about a typical Gaussian curve of random distribution, namely one having a standard deviation of

*; this curve has been drawn in the diagram, and, as will be seen, the agreement is good, considering the smallness of the sample Fic. 2. Grapao or Seconp Moprt Process ILLUSTRATING THE STATISTICAL Number of pendulums found on left of median position at successive epochs. (Reproduced from A. J. Lotka, Two Models in Statistical Mechanics, Am. Math. Monthly, vol. 31, 1924, p. 124.) (412 observations, extending over 41.2 seconds, out of a total period of 7385 years). Thus for long stretches of time the periodicity of the motion of the system of pendulums is very effectively masked under an aspect of ‘‘chance.”

These simple models illustrate very clearly how the seeming conflict between the periodicity of all mechanical motions and the apparently one-sided course of events, directed toward one definite end state, is resolved. The actual process of isothermal gaseous diffusion is, in fact, periodic, but with a period so long that humanly speaking, the return to the initial state never occurs at all. For all stretches of time that can have any real significance in human thought (and this includes the vast historical ranges of all geology and astronomy), it may therefore be said in a certain sense that evo-

lution proceeds, in all but a vanishingly small class of exceptional cases, from less probable states (e.g., uneven distribution of the 50 black balls in the two urns) to more probable states, tending ultimately toward a most probable state. This statement cannot however be allowed to pass without a word of caution. It is mean- SO9_ 6a 4 ie Oy eee GS 89 Fig. 3. Frequpncy DIAGRAM FOR THE DEVIATIONS FROM THE MEAN Abscissae represent deviations from the mean (13) in number of pendulums on left of median; ordinates represent corresponding frequencies, among the observations recorded in figure 2. (Reproduced from A. J. Lotka,

ingless unless the characteristic with regard to which probability is reckoned is explicitly or implicitly indicated. Probability is essentially a matter of classification. An improbable event is one that is a member of a small class, and whether it is so or not depends, clearly, on our system of classification. For this reason the broad statement which has sometimes been made,” that the direction of evolution is from less probable to more probable states, is not only inadequate, but is really meaningless. It is indefinite in failing to specify with regard to what characteristic probability is to be reckle oned; and it is incomplete in failing to call attention to the fundamentally important connection between the particular probabilities in question and available energy.

Another point, which has not hitherto perhaps received its deserved attention is clearly brought out by the two models deviously, if we use our visual discrimination in selecting the balls drawn from the boxes (instead of drawing blindly), we can easily bring it about that in short order all the black balls are back in box A. Thus a process may be reversible or not, according to the means that are naturally available or arbitrarily permitted in operating upon the system under consideration; somewhat as the trisection of an angle is or is not an impossible geometric construc- } tion, according as we are or are not forbidden the use of instruments other than ruler and compass. In the case of molecular aggregates this fact has long been duly appreciated, having been first pointed out by Clerk Maxwell, who remarked that a demon capable of dealing with individual molecules would be able to cheat the second law of thermodynamics. But it seems to have been pretty generally overlooked that the relative character of irreversi- / bility has an important significance in certain natural processes taking place on a macroscopic scale. In point of fact this is a matter whose importance in the world of living organisms can hardly

be rated too high. For there are certain diffusive processes going on in nature which, from the standpoint of thermodynamics, are , not of the irreversible type; but which might as well be, so far as any benefit derived from their reversibility by the organism (and, in particular, by man) is concerned. If I should be the fortunate possessor of a pound of gold dust, and some malicious person should take it and scatter it far and wide, so that it became hopelessly diluted with dust and refuse, it would be a small comfort to me to know that it were merely mechanically commingled with such foreign matters, that it had not irreversibly undergone solution or chemical transformation, and that therefore it could theoretically be recovered without the expenditure of work. In practice its recovery might entail the expenditure of far more energy than if the gold were present in reasonably concentrated solution. The point is that in practice I am restricted to operations in bulk upon reasonably large quantities, in reasonably concentrated form, otherwise the theoretical ideal of recovery without work is very far from being attained. And with this restriction placed upon my operations, certain processes acquire an irreversibility which they do not possess apart from that restriction. The illustration of the pound of gold has the advantage of simplicity and cogency. But if by any chance it has conveyed the impression that only in peculiar and far-fetched cases does this kind of irreversibility enter into play, then it is an unfortunate example indeed, for nothing could | be farther from the truth. The fact is that nature abounds in | just such dissipative processes as the scattering to the four winds, to utter inutility, of materials of the highest importance to life; and _ one of the central problems which the organism has to solve in the _ struggle for existence, is the reconcentration, into his immediate ) environment and into his body, of valuable materials that have _ become scattered by agencies beyond his control.

It is not the least of the triumphs which have made man the lord of creation, that he has learnt, beyond all comparison more effectively than any of his competitors, to carry out this process of reconcentration to satisfy his needs. So a fleet of ships, year by year, bear a burden of saltpeter from Chile to all civilized countries, to balance the losses from our depleted fields. So, in his most recent technical achievement, man has learnt to draw from the air a supply that will continue unfailing, long after the Chilean nitre beds are exhausted. The fact is, in dealing with the physics of such macroscopically irreversible effects, it will ultimately be necessary to develop a method of mathematical analysis that shall be competent to distinguish and handle not only the extremes—the case of a primitive organism that can deal only in the gross, without intelligent or

other discrimination, with the matter and situations presented to it; and an ideally perfect organism, that should expend just the minimum of effort, directed with absolute precision toward the attainment of its ends. A method must be devised that shall duly take account of, and use as a fundamental datum for its deductions, the particular character, the particular degree of perfection of the mechanical and psychic equipment or organization by which each organism reacts more or less selectively upon its environment. We shall have occasion to refer to this matter again in greater detail in a later section. But here it is well to note that our two models are suggestive also with regard to this aspect of the subject. For it appears at first sight as if there were a fundamental difference in character between the first and the second model, since it is essential for the operation of the urn model that the drawing be done blindly, so as to give chance a part in the process, as we would say; whereas the pendulum model we operate with our eyes open, apparently in full consciousness of what is going on. Chance seems to play no part here, the system is mechanically determinate. But there is a blindness which is not of the eye, and there is a vision that surpasses optical vision. The same struggle for existence which has developed in man the organ of sight, to depict for him the external world, to furnish him with a map on which to base his plan of campaign, has also, in latter days, developed his internal vision, whereby he extends his world-picture beyond the powers of the bodily eye.

It is immaterial by which process his map is drawn—its function is the same; whether I peep into the urn and manipulate the drafts by the light of my eyes; or whether, in the light of my knowledge of mechanics, I adjust the pendulums to equal Jengths and phases; or again, whether, in the more serious affairs of life I employ these same faculties to diverse ends, | the effect is the same: In greater measure or less these organs and | faculties emancipate me from the bonds of the fortuitous and make | me a controller of events. Their function is to substitute choice for chance, to introduce aimed collisions in place of random encounters. | Origin of Subjective Sense of Direction-in-Time. The failure of the differential equations of dynamics to discriminate between ¢ and —t raises the question as to the physical significance and origin of our subjective conviction of a fundamental difference between the forward and the backward direction in time,—a con-

viction that is intimately bound up with the concept of evolution, for, whatever may ultimately be found to be the law of evolution, it is plain that no trend of any kind can be defined or even described without reference to a favored direction in time. One view which suggests itself is that this conviction is our subjective appreciation of the trend from less probable to more probable states recognized in statistical mechanics. But this does not seem very satisfying, for we somehow feel that our conviction must rest on something more fundamental than this somewhat accidental circumstance, which, as the models described clearly show, is fundamentally incompetent to distinguish between the forward and the backward direction in time. For the peak in figure 1, for example, may indifferently be traversed from left to right or vice versa, it presents the same general character in either sense.

Another alternative is to suppose that the differential equations of dynamics, as formulated by us today, are either an incorrect, or else an incomplete statement of facts. The latter view is, indeed, upon reflection, found to have a certain warrant. For the differential equations of motion alone do not fully determine the actual course of events; this depends further on the value of certain arbitrary constants of integration; or, to speak in terms of physical entities, upon the initial velocities of the particle of which the system is composed. Strictly speaking it is only when the initial velocities are zero, that the equations of motion, considered in their totality, are indifferent to the substitution ¢t’ =.—¢. From this point of view our sense of the forward direction in time would appear as our subjective appreciation of the fact that, once a material system has been started on a certain course, with certain initial velocities, there then remains no further freedom; its history must continue to unfold in the direction determined by the initial veloc-

It seems, however, that it is not with this perfectly general type of irreversibility of the course of events that we are chiefly concerned in the study of evolution. The concept of evolution, according to the analysis which has been made of it in preceding pages, applies principally, if not exclusively, to systems that outwardly at least affect the aperiodic habit, systems that do not return periodically to their initial state, but show a definite trend, whereby yesterday and tomorrow are never alike, and differ more-

over in some definite and characteristic fashion, even though we may not be fully competent, at the present epoch of science, to specify exactly wherein lies the characteristic difference.’ Inadequacy of Thermodynamic Method. Our reflections so far have linked the fundamental problem of the direction, the trend of evolution, with the disciplines of thermodynamics and statistical mechanics. From this point of view the direction of evolution is identified with the direction of the unfolding of irreversible proc- ¥

esses, the direction of increase of entropy (in thermodynamics) or of increasing probability (in statistical mechanics). A certain mental satisfaction may be derived from this conclusion. It gives us, in principle at least, an answer to our question “Quo vadis?”’ But practically the answer is very inadequate. If the conclusions, the methods of thermodynamics, or of statistical mechanics, are to be applied to a concrete case, the data of the problem must be presented in a very particular form. So long as we deal with volumes, pressures, temperatures, etc., our thermodynamics serve us well. But the variables in terms of which we find it convenient to define the state of biological (life-bearing) systems are other than these. We may have little doubt that the principles of thermodynamics or of statistical mechanics do actually control the processes occurring in systems in the course of organic evolution. But if we seek to make concrete application we find that the systems under consideration are far too complicated to yield fruitfully to thermodynamic reasoning; and such phases of statistical mechanics as relate to aggregation of atoms or mole-

cules, seem no better adapted for the task. To attempt applica- | tion of these methods to the prime problems of organic evolution is much like attempting to study the habits of an elephant by means of amicroscope. It is not that the substance of the elephant is inherently unfitted to be viewed with the microscope; the instrument is ill adapted to the scale of the object and the investigation. It would seem, then, that what is needed is an altogether new instrument; one that shall envisage the units of a biological population as the established statistical mechanics envisage molecules, atoms and electrons; that shall deal with such average effects as

3 Perhaps the objective interpretation of our subjective sense of direction in time must be sought in quantum mechanics. Cf. A. J. Lotka, loc. cit., p, 126, and, W. S. Franklin, Science, 1924, vol. 60, p. 258. population density, population pressure, and the like, after the manner in which thermodynamics deal with the average effects of gas concentration, gas pressures, etc.; that shall accept its problems in terms of common biological data, as thermodynamics accepts problems stated in terms of physical data; and that shall give the answer to the problem in the terms in which it was presented. What is needed, in brief, is something of the nature of what has been termed “Allgemeine Zustandslehre,’’ a general method or Theory of State.

It is somewhat along these lines that the system now to be sketched is conceived. Toutes ces choses ne peuvent se determiner surement que par des mesures précises que nous chercherons plus tard; mais auparavant il fallait au moins sentir le besoin de les chercher.—J. B. Biot. It now behooves us to establish, with respect to the problem of evolution, a viewpoint, a perspective, a method of approach, which has hitherto received its principal development and application outside the boundaries of biological science. Such prior development and applications, however extraneous to our chief line of interest here, may well serve us in our present interrogations, since we shall be in a position to profit by the precedents established in methods, in conclusions, and, most particularly, in habit of thought.

This perspective is that which contemplates an evolving system as an aggregation of numbered or measured components of several specified kinds, and which observes and enregisters the history of that system as a record of progressive changes taking place in the distribution, among those components, of the material of which the system is built up. It is thus that physical chemistry views the progressive changes in a system comprising several chemical species, that is to say elements, compounds, phases, etc. It describes the system by enumerating these components, by stating their character and extent (mass); and by further indicating the values of certain quantities or parameters, such as volume or pressure, temperature, etc., which, together with the masses of the components, are found experimentally to be both necessary and sufficient, for the purposes in view, to define the state of the system. With the instantaneous state of the system thus defined, physical chemistry investigates by observation and by deductive reasoning (theory) the history, the evolution of the system, and gives analytical expression to that history, by establishing relations, or equations, between the variables defining these states (after the manner set forth above), and the time.

It is commonly found that these fundamental equations assume the simplest, the most perspicuous form, when they are written relative to rates of change of the state of the system, rather than \ relative to this state itself. That is to say, it is found that the expressions for the rate of increase in mass, the velocity of growth, _ of the several components, are simpler, more primitive in form, than | the expressions giving directly the mass of each component as a

function of the time. In the language of the calculus, the differential equations display a certain simplicity in form, and are therefore, in the handling of the theory at least, taken as the starting point, from which the equations relating to the progressive states themselves, as functions of the time, are then derived by integration.! So, for example, a simple system may be defined as comprising 4 gram-molecules of hydrogen, 2 gram-molecules of oxygen, and 100 gram-molecules of steam, at one atmosphere pressure, and at 1800°C. The fundamental relation expressing the law of evolution, the historical pattern, of the system, is in this case given by the law of mass action:

where v is the volume, m, is the mass of steam, m2 the mass of hydrogen, and ms, the mass of oxygen (all expressed in gram-molecules). The coefficients k;, kz, are functions of the temperature, or, for a given temperature, are characteristic constants of the reaction. We are not, here, interested in the particular form of the law of mass action. What does interest us is the general form of the equation (1). It states that the rate of increase in mass, the velocity of growth of one component, steam (mass mj), is a function of the masses m2, ms, of the other components, as well as of the mass m, itself, and, besides, of the parameters v (volume) and 7 (temperature), the latter being contained in the coefficients k, k,. This statement, in its more general form, is written, according to established notation?

‘In experimental observation usually (though not always) the reverse attitude is adopted. * For the benefit of the non-mathematical reader it may here be explained that equation (2) is merely a short-hand expression, so to speak, of the simple ‘ Re ci! statement: The rate of increase of mass with time of the component S;, is a function of, or is determined by, the masses mi, M2, M3, of the components Si, S2, Sz, as well as by the volume v and the temperature 7. Precisely similar is the construction to be placed on the equations (3).

; Now it is this habit of thought, expressed in equation (2), that is to be transplanted into the contemplation of problems of evolution in general, and organic evolution in particular; this point of view, this perspective, which regards evolution as a process of redistribution of matter among the several components of a system, under specified conditions.* Having thus passed from the specific to the general—from the case of physico-chemical systems to a general formulation—we now retrace our steps to the particular, but in a new direction. We now contemplate the kind of systems that form the object of study of the biological sciences.

With the outlook gained in our preceding reflections we envisage the life-bearing system, in the progress of evolution, as an assembly of a number of components: Biological species; collections or aggregations of certain inorganic materials such as water, oxygen, carbon dioxide, nitrogen, free and in various combinations, phosphorus, sulphur, ete. These components are placed in various relations of mutual interaction under specific conditions of area, topography, climate, etc.

Under these conditions each may grow, decay, or maintain equilibrium. In general the rate of growth Uo any one of these components will depend upon, will be a function of, the abundance in which it and each of the others is presented; this rate of growth will also be a function of the topography,‘ climate, ete. If these latter features are defined in terms of a set_of parameters P,P... . Pj, we may write, in the same sense as equation (2) 4 Terrestial species have an essentially two-dimensional distribution, so that area functions here in a manner somewhat analogous to that in which volume enters into physico-chemical relations. Aquatic life, with its threedimensional sphere of activity, is enacted in systems whose extension is described in terms of volume. More detailed topographic parameters may be required to define in sufficient completeness the configuration, the structure of these systems. The absence of such detail from the more familiar formulations of chemical dynamics is due to the purely accidental circumstance that the systems commonly dealt with in that branch of science are either homogeneous, or of comparatively simple heterogeneous structure.

Ot Fy (Xu Xe Xnj Py Psy - - - Pa) “te FP, (Xi, Xe, At Xn} Ps P2, cake Pj) (3) WP = Fa (Xs Xx. -Xnj Py Py +. Pi In general there will be n such equations, one for each of the n components. The purport of these equations (3) remains uncertain so long as the components (e.g., biological species, etc.) S; S2 . . . Sy, are undefined. What definitions we may adopt for these is, in accordance with the principles discussed anteriorly, a matter for arbitrary disposition; though we may be guided in our choice by considerations of expediency. These may advise different policies from case to case, according to the particular phase of the problem taken in view. The conclusions reached will, of course, depend upon the particular definitions chosen. This is as it should be; the conclusions apply to the components as defined, and, in general, to no other. This seems clear enough, but if any further exposition is needed, it will be found in examples shortly to be considered.

Intra-Group Evolution. While the precise definition of the components S; S. . . . S, may, and indeed must, be held over for determination as each separate phase of the general problem is singled out for treatment, yet there is one class of cases regarding which it is appropriate to set forth certain reflections at this juncture. It may have been observed that so far nothing has been said regarding one aspect of organic evolution which, in the history of the subject in the past, and in the minds and writings of its exponents and students today, occupies a dominating position— namely, the relation of evolution to the modification of species with the passage of time, and, in particular, to the origin of new species.